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12345 [234]
3 years ago
9

Help plzzzzzzzzzzzzzz ​

Mathematics
1 answer:
Viefleur [7K]3 years ago
6 0

Answer:

∠E

Step-by-step explanation:

∠E is between PE and EN

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Problem 4: Let F = (2z + 2)k be the flow field. Answer the following to verify the divergence theorem: a) Use definition to find
Viktor [21]

Given that you mention the divergence theorem, and that part (b) is asking you to find the downward flux through the disk x^2+y^2\le3, I think it's same to assume that the hemisphere referred to in part (a) is the upper half of the sphere x^2+y^2+z^2=3.

a. Let C denote the hemispherical <u>c</u>ap z=\sqrt{3-x^2-y^2}, parameterized by

\vec r(u,v)=\sqrt3\cos u\sin v\,\vec\imath+\sqrt3\sin u\sin v\,\vec\jmath+\sqrt3\cos v\,\vec k

with 0\le u\le2\pi and 0\le v\le\frac\pi2. Take the normal vector to C to be

\vec r_v\times\vec r_u=3\cos u\sin^2v\,\vec\imath+3\sin u\sin^2v\,\vec\jmath+3\sin v\cos v\,\vec k

Then the upward flux of \vec F=(2z+2)\,\vec k through C is

\displaystyle\iint_C\vec F\cdot\mathrm d\vec S=\int_0^{2\pi}\int_0^{\pi/2}((2\sqrt3\cos v+2)\,\vec k)\cdot(\vec r_v\times\vec r_u)\,\mathrm dv\,\mathrm du

\displaystyle=3\int_0^{2\pi}\int_0^{\pi/2}\sin2v(\sqrt3\cos v+1)\,\mathrm dv\,\mathrm du

=\boxed{2(3+2\sqrt3)\pi}

b. Let D be the disk that closes off the hemisphere C, parameterized by

\vec s(u,v)=u\cos v\,\vec\imath+u\sin v\,\vec\jmath

with 0\le u\le\sqrt3 and 0\le v\le2\pi. Take the normal to D to be

\vec s_v\times\vec s_u=-u\,\vec k

Then the downward flux of \vec F through D is

\displaystyle\int_0^{2\pi}\int_0^{\sqrt3}(2\,\vec k)\cdot(\vec s_v\times\vec s_u)\,\mathrm du\,\mathrm dv=-2\int_0^{2\pi}\int_0^{\sqrt3}u\,\mathrm du\,\mathrm dv

=\boxed{-6\pi}

c. The net flux is then \boxed{4\sqrt3\pi}.

d. By the divergence theorem, the flux of \vec F across the closed hemisphere H with boundary C\cup D is equal to the integral of \mathrm{div}\vec F over its interior:

\displaystyle\iint_{C\cup D}\vec F\cdot\mathrm d\vec S=\iiint_H\mathrm{div}\vec F\,\mathrm dV

We have

\mathrm{div}\vec F=\dfrac{\partial(2z+2)}{\partial z}=2

so the volume integral is

2\displaystyle\iiint_H\mathrm dV

which is 2 times the volume of the hemisphere H, so that the net flux is \boxed{4\sqrt3\pi}. Just to confirm, we could compute the integral in spherical coordinates:

\displaystyle2\int_0^{\pi/2}\int_0^{2\pi}\int_0^{\sqrt3}\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi=4\sqrt3\pi

4 0
3 years ago
Can anyone help me with the problem in the picture?
torisob [31]
X = 45.

Since opposite exterior angles are congruent, you can set the 2 values equal to each other. When you solve for x, you should get 45.
3 0
3 years ago
. Find the surface area of a sphere with diameter 30 m. Leave your answer in terms of
OlgaM077 [116]

The sphere has surface are of 900π m².

Step-by-step explanation:

Given,

Diameter = d = 30m

Radius = \frac{d}{2} =\frac{30}{2} = 15m

Surface area of sphere = 4\pi\ r^2

Surface\ area = 4*\pi *(15)^2\\Surface\ area = 4*\pi * 225\\Surface\ area= 990\pi

The sphere has surface are of 900π m².

Keywords: Surface area, multiplication

Learn more about spheres at:

  • brainly.com/question/2367554
  • brainly.com/question/2821386

#LearnwithBrainly

3 0
3 years ago
Solve equation 4x=20
Romashka [77]

4x=20

4/4x=20/4

x=5

u want to isolate the variable to solve these equations

6 0
3 years ago
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The output is 11 more than the input<br> A: y=x+11<br> B: y=11x<br> C: y=x/11<br> D: x=y=11
Alchen [17]
The answer would be A
5 0
3 years ago
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